 Solution

Integrate the ordinary differential equation across one step, using the fundamental theorem of calculus:The midpoint quadrature rule approximates the integral by $hf(x_n+h/2,y(x_n+h/2))$. Approximate the unknown midpoint state by the midpoint value of linear interpolation between the endpoint states:Combining these approximations gives the implicit midpoint ruleIt is an implicit time-stepping method because its new state occurs inside the evaluation of $f$. An equivalent derivation uses a straight-line approximation on the step and imposes the ordinary differential equation at its midpoint, a one-stage collocation method.